Abstract:
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Spatial data are assumed to possess properties of stationarity or non-stationarity via a kernel fitted to a covariance matrix. A primary workhorse of stationary spatial statistics is Gaussian maximum log-likelihood estimation (MLE), whose central data structure is a dense, symmetric positive definite covariance matrix of the dimension of the number of correlated observations. In this contribution, we reduce the precision of weakly correlated locations to single- or half-precision based on distance. We thus exploit mathematical structure to migrate MLE to a three-precision approximation that takes advantage of contemporary architectures offering extremely fast linear algebra operations for reduced precision. We also add another level of approximation by mixing different precision with tile low-rank approximation to gain more performance. Finally, we assess the accuracy of our proposed implementation on large-scale using four supercomputers with different architectures, i.e., HAWK-HLRS (AMD CPUs), Shaheen-II-KAUST (Intel CPUs), Summit-ORNL (NVIDIA GPUs), and Fujitsu A64FX (Fugaku-Riken). The experiments have been performed on up to 12M covariance matrix using synthetic and real data.
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